419 research outputs found
Polynomiality of shifted Plancherel averages and content evaluations
The shifted Plancherel measure is a natural probability measure on strict
partitions. We prove a polynomiality property for the averages of the shifted
Plancherel measure. As an application, we give alternative proofs of some
content evaluation formulas, obtained by Han and Xiong very recently. Our main
tool is factorial Schur -functions.Comment: 23 page
Hyperdeterminantal expressions for Jack functions of rectangular shapes
We derive a Jacobi-Trudi type formula for Jack functions of rectangular
shapes. In this formula, we make use of a hyperdeterminant, which is Cayley's
simple generalization of the determinant. In addition, after developing the
general theory of hyperdeterminants, we give summation formulae for Schur
functions involving hyperdeterminants, and evaluate Toeplitz-type
hyperdeterminants by using Jack function theory.Comment: 18 pages. The title of 3rd version is ''Hyperdeterminant expressions
...''. The title of 1st and 2nd versions was ``Summation formulas for Schur
functions involving hyperdeterminants'
ABJ Fractional Brane from ABJM Wilson Loop
We present a new Fermi gas formalism for the ABJ matrix model. This
formulation identifies the effect of the fractional M2-brane in the ABJ matrix
model as that of a composite Wilson loop operator in the corresponding ABJM
matrix model. Using this formalism, we study the phase part of the ABJ
partition function numerically and find a simple expression for it. We further
compute a few exact values of the partition function at some coupling
constants. Fitting these exact values against the expected form of the grand
potential, we can determine the grand potential with exact coefficients. The
results at various coupling constants enable us to conjecture an explicit form
of the grand potential for general coupling constants. The part of the
conjectured grand potential from the perturbative sum, worldsheet instantons
and bound states is regarded as a natural generalization of that in the ABJM
matrix model, though the membrane instanton part contains a new contribution.Comment: 28 pages, 5 eps figures, v3: typos corrected and references added,
version to appear in JHE
Linear versus spin: representation theory of the symmetric groups
We relate the linear asymptotic representation theory of the symmetric groups
to its spin counterpart. In particular, we give explicit formulas which express
the normalized irreducible spin characters evaluated on a strict partition
with analogous normalized linear characters evaluated on the double
partition . We also relate some natural filtration on the usual
(linear) Kerov-Olshanski algebra of polynomial functions on the set of Young
diagrams with its spin counterpart. Finally, we give a spin counterpart to
Stanley formula for the characters of the symmetric groups.Comment: 41 pages. Version 2: new text about non-oriented (but orientable)
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